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Theorem eqsb3 2942
Description: Substitution applied to an atomic wff (class version of equsb3 2110). (Contributed by Rodolfo Medina, 28-Apr-2010.)
Assertion
Ref Expression
eqsb3 ([𝑦 / 𝑥]𝑥 = 𝐴𝑦 = 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem eqsb3
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2828 . 2 (𝑥 = 𝑤 → (𝑥 = 𝐴𝑤 = 𝐴))
2 eqeq1 2828 . 2 (𝑤 = 𝑦 → (𝑤 = 𝐴𝑦 = 𝐴))
31, 2sbievw2 2108 1 ([𝑦 / 𝑥]𝑥 = 𝐴𝑦 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  [wsb 2070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-cleq 2817
This theorem is referenced by:  pm13.183  3644  eqsbc3  3803
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